Our strategy to count Euler tours is as follows: Given a bounded treewidth graph \(G\) ,
we count the number of Euler tours of \(G\)  by counting the number of Hamiltonian tours of the
line graph of \(G\) , \(L(G)\) . In general, there is no bijection between these two quantities, but
we show that \(G\)  can be modified to obtain \(G^{\prime }\)  (\(tw(G^{\prime }) \le tw(G) + 3\) ) such that \(G\) , \(G^{\prime }\)  have
the same number of Eulerian tours, which equals the number of Hamiltonian tours of \(L(G^{\prime })\) .
Henceforth, we will be primarily interested in line graphs of bounded treewidth
graphs. It is known that such graphs are of bounded clique-width [1]}
